# Sliding mode control of the Hodgkin-Huxley mathematical model

## TL;DR

This paper develops a sliding mode control strategy for the Hodgkin-Huxley neuronal model, using an external current to steer the membrane potential to a desired state and maintain it there, supported by mathematical proofs and simulations.

## Contribution

It introduces a novel control approach for the Hodgkin-Huxley model employing relay-based feedback to achieve finite-time reaching and sliding on a target manifold.

## Key findings

- Proves existence of sliding mode in the Hodgkin-Huxley system.
- Provides a method to determine the time to reach the target manifold.
- Demonstrates effectiveness through numerical simulations.

## Abstract

In this paper we deal with a feedback control design for the action potential of a neuronal membrane in relation with the non-linear dynamics of the Hodgkin-Huxley mathematical model. More exactly, by using an external current as a control expressed by a relay graph in the equation of the potential, we aim at forcing it to reach a certain manifold in finite time and to slide on it after that. From the mathematical point of view we solve a system involving a parabolic differential inclusion and three nonlinear differential equations via an approximating technique and a fixed point result. The existence of the sliding mode and the determination of the time at which the potential reaches the prescribed manifold is proved by a maximum principle argument. Numerical simulations are presented.